next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Macaulay2Doc :: CoherentSheaf ^** ZZ

CoherentSheaf ^** ZZ -- tensor power

Synopsis

Description

The second symmetric power of the canonical sheaf of the rational quartic:
i1 : R = QQ[a..d];
i2 : I = monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of R
i3 : X = variety I

o3 = X

o3 : ProjectiveVariety
i4 : KX = sheaf(Ext^1(I,R^{-4}) ** ring X)

o4 = cokernel {1} | c 0 -d 0 -b |
              {1} | b c 0  a 0  |
              {1} | 0 d c  b a  |

                                         3
o4 : coherent sheaf on X, quotient of OO  (-1)
                                        X
i5 : K2 = KX^**2

o5 = cokernel {2} | c 0 -d 0 -b 0 0 0  0 0  0 0 0  0 0  c 0 -d 0 -b 0 0 0  0 0  0 0 0  0 0  |
              {2} | b c 0  a 0  0 0 0  0 0  0 0 0  0 0  0 0 0  0 0  c 0 -d 0 -b 0 0 0  0 0  |
              {2} | 0 d c  b a  0 0 0  0 0  0 0 0  0 0  0 0 0  0 0  0 0 0  0 0  c 0 -d 0 -b |
              {2} | 0 0 0  0 0  c 0 -d 0 -b 0 0 0  0 0  b c 0  a 0  0 0 0  0 0  0 0 0  0 0  |
              {2} | 0 0 0  0 0  b c 0  a 0  0 0 0  0 0  0 0 0  0 0  b c 0  a 0  0 0 0  0 0  |
              {2} | 0 0 0  0 0  0 d c  b a  0 0 0  0 0  0 0 0  0 0  0 0 0  0 0  b c 0  a 0  |
              {2} | 0 0 0  0 0  0 0 0  0 0  c 0 -d 0 -b 0 d c  b a  0 0 0  0 0  0 0 0  0 0  |
              {2} | 0 0 0  0 0  0 0 0  0 0  b c 0  a 0  0 0 0  0 0  0 d c  b a  0 0 0  0 0  |
              {2} | 0 0 0  0 0  0 0 0  0 0  0 d c  b a  0 0 0  0 0  0 0 0  0 0  0 d c  b a  |

                                         9
o5 : coherent sheaf on X, quotient of OO  (-2)
                                        X
i6 : prune K2

o6 = cokernel {1} | c2 bd ac b2 |
              {2} | -d -c -b -a |

                                         1           1
o6 : coherent sheaf on X, quotient of OO  (-1) ++ OO  (-2)
                                        X           X
Notice that the resulting sheaf is not always presented in the most economical manner. Use prune to improve the presentation.

See also